# $n$-product-periodic topological spaces

We call an topological space $$(X,\tau)$$ $$n$$-product-periodic for an integer $$n\geq 3$$ if $$\prod_{i=1}^n X \cong X$$ but for all integers $$k$$ with $$2\leq k\leq n-1$$ we have $$\prod_{i=1}^k X \not\cong X$$.

Is there an integer $$n\geq 3$$ such that there is an $$n$$-product-periodic space, and is there an integer $$m\geq 3$$ such that there is no $$m$$-product-periodic space?

Garrett Ervin's answer to When is $$A$$ isomorphic to $$A^3$$? mentions also some results on topological spaces. (Although the question was originally about abelian groups.)

The results mentioned there seem to answer your question - although I do hope that somebody can provide a more elementary solution. (The result from these papers have several additional requirements on the space $$X$$.)

In particular, the linked answer mentions the paper A. Orsatti and N. Rodino, Homeomorphisms between finite powers of topological spaces, Topology Appl. 23 (1986), no. 3, 271--277; MR858335, Zbl 0603.54009.

Let $$\lambda$$ be an infinite cardinal number. It is proved that, for each positive integer $$r$$, there exists a compact connected homogeneous topological space $$X$$ of weight $$\lambda$$ such that $$X^n$$ is homeomorphic to $$X^m$$ iff $$n\equiv m \pmod r$$. The cardinality of the set of homeomorphism classes of compact connected homogeneous spaces with this property is exactly $$2^\lambda$$. Moreover every completely regular space of weight $$\lambda$$ is embeddable in a space of this type.

Another paper with related results of this type is Věra Trnková: Products of metric, uniform and topological spaces. Commentationes Mathematicae Universitatis Carolinae, vol. 31 (1990), issue 1, pp. 167-180; MR1056184, Zbl 0696.54009.

For every triple of natural number $$a$$, $$b$$, $$c$$ there exists a metric space $$X$$, the $$m$$-th power and the $$n$$-th power of which are

• homeomorphic iff $$m\equiv n \pmod a$$
• uniformly homeomorphic iff $$m\equiv n \pmod {ab}$$
• isometric iff $$m\equiv n \pmod {abc}$$

This is a consequence of the main theorem proved in the present paper, where simultaneous representations of commutative semigroups by the products of metric, uniform and topological spaces are investigated.